Global well-posedness of the Majda-Biello system in the resonant case on the real line

TL;DR

This paper proves global well-posedness for the Majda-Biello system at a critical Sobolev regularity on the real line by introducing a novel dual-operator I-method tailored to its resonant structure.

Contribution

It introduces a dual-operator I-method to handle resonance effects, extending global well-posedness results to the critical Sobolev index for the first time.

Findings

Established global well-posedness for s in [3/4, 1)

Refined the I-method with dual operators for resonant systems

Controlled Sobolev norms via almost conserved modified energies

Abstract

We study the Cauchy problem for the following Majda-Biello system in the case $\alpha=4$, where the resonance effect is the most significant, on the real line. \[ \left\{ \begin{array}{rcl} u_{t} + u_{xxx} & = & - v v_x, v_{t} + \alpha v_{xxx} & = & - (uv)_{x}, (u,v)|_{t=0} & = & (u_0,v_0) \in H^{s}(\mathbb{R}) \times H^{s}(\mathbb{R}), \end{array} \right. \quad x \in \mathbb{R}, \, t \in \mathbb{R}. \] For Sobolev regularity $s\in[\frac34, 1)$, we establish global well-posedness by refining the I-method. Previously, the critical index for local well-posedness was known to be $\frac34$, while global well-posedness was only obtained for $s\geq 1$. Our global well-posedness result bridges the gap and matches the threshold in the local theory. The main novelty of our approach is to introduce a pair of distinct $I$-operators, tailored to the resonant structure of the Majda-Biello system with $\alpha=4$. This dual-operator framework allows for pointwise control of the multipliers in the modified energies constructed via the multilinear correction technique. These modified energies are almost conserved and provide effective control over the Sobolev norm of the solution for all time. This new approach has potential applications to other coupled dispersive systems exhibiting strong resonant interactions.